Higher-Dimensional Theories Generalizing Kaluza’s

11 Higher-Dimensional Theories Generalizing Kaluza’s

11.1 5-dimensional theories: Jordan–Thiry theory

In January 1950, Yves Thiry submitted a thesis to the faculty of science of Paris University with the title “Mathematical study of the equations of a unitary theory with fifteen field variables”. He exuberantly thanked his “master and friend Lichnerowicz” who obviously had initiated the work. Unfortunately, “Jordan and his school” had “obtained almost at the same time like us the equations which we will give in Chapter II. We had no knowledge about this except at a very late stage, and it is only recently that we could correspond with Jordan. He was so friendly as to send us his publications which we could not have procured otherwise.”²²⁷ ([606 *], p. 6) In fact, it was A. Lichnerowicz, then at the university of Strasbourg, who had written to W. Pauli and asked for “Jordan’s original paper” (cf. letter of W. Pauli to P. Jordan of 23. 3. 1948 in [489 *], p. 516). According to Pauli:

“Lichnerowicz is a pure mathematician who is occupied with the integration of Einstein’s field equations. One of his students, Ives Thiry now has looked into the (not mutilated) Kaluza-theory (with $g55$ ) and, so I believe, has simplified very much the calculational technique.”²²⁸

Since 1947/48, first together with A. Lichnérowicz, Thiry had published short notes about Kaluza’s theory in Comptes Rendus [381 *, 604 *, 605 ]. On 19 January 1948,²²⁹ he sent one of his notes in Comptes Rendus to Albert Einstein [604 *]. In his publication preceding Lichnerowicz’ letter to Pauli, Thiry had not yet given a physical interpretation of the scalar field [604 ]. Around that time, the interest in five dimensional relativity seems to have risen; we already have met P. G. Bergmann’s paper of 1948 [21 ].²³⁰ C. V. Jonsson, a student of O. Klein in Stockholm, also wrote a long paper about the theory’s field equations and their linear approximation. He included the scalar field and dropped the cylinder condition.²³¹ He then quantized the free field of this linear regime. [313 ]. Thiry’s interest in Kaluza’s theory was of mathematical nature:

“As to unitary field theories, it seems that their mathematical study has been quite neglected […]. We thought it useful to try a systematic mathematical study of a unitary field theory, and to find out whether such a theory is able to present the same coherence like general relativity.” ([606 *], p. 3)²³²

Thiry’s thesis laid out in three chapters the conceptual background of a 5-dimensional theory, the setting up and study of the field equations by help of Cartan’s differential calculus, and results on regular solutions of the theory’s field equations. Unlike in the approach by Einstein & Bergmann (cf. Section 3.2), throughout, the cylinder condition $g αβ,4 = 0$ is upheld. Here, he used an argument from physics: no physical phenomena furnishes evidence for the existence of a fifth dimension ([606 *], p. 39). His access to 5-dimensional space made use of the fact that the equations of motion of charged particles are geodesics in Finsler geometry; for each value of $e- m$ (charge over mass) another Finsler space is needed with the metric:

He then showed that a 5-dimensional Riemannian space could house all these geodesics.²³³

“The introduction of a fifth coordinate […] thus shall justify itself by the fact that it imparts the role of geodesics to the trajectories of charged particles which they lost in space-time […]”²³⁴

In the third chapter, Thiry aimed at showing that his unitary field theory possessed the same mathematical coherence with regard to its global aspects as general relativity. By partially using methods developed by Lichnerowicz, he proved theorems on the global regularity of solutions such as: “A unitary field with normal asymptotic behavior (i.e., tending uniformly to Minkowski space at spatial infinity) cannot be regular everywhere.” Thiry compared the proofs on the existence and regularity of solutions in O. Klein’s version of Kaluza’s theory and in his generalization and found them much simpler in his theory. These results are of a different nature than what Jordan had achieved; they are new and mathematically exact.

Yet Jordan was not in a hurry to read Thiry’s thesis; he wrote to Pauli:

“By the way, in his thèse published in 1951, Thiry has studied systematically and extensively the theory with variable gravitational constant; […] I received it only after my book appeared and, at present, I have not read it very closely. It thus is not really clear to me whether it contains interesting novelties.” ([490 *], p. 799/800)²³⁵

To Pauli, Thiry’s global theorems might not have been “interesting novelties”, because in his corresponding paper with Einstein on the non-regularity of solutions, the proof had been independent of the dimension of space [177 ]. Pauli, at first, also did not read Thiry’s thèse, but responded arrogantly:

“The Thèses by Thiry are laying on my desk; however they are so appallingly thick (do not contain a reasonable abstract) such that it is so much simpler to not open the book and reflect about what must be inside.” (W. Pauli to P. Jordan 8. 6. 1953, [491 *], p. 176)²³⁶

Somewhat later, Pauli corrected himself and wrote to Jordan that in the preparation for a course “he nevertheless had read around in Thiry’s Thèse” W. Pauli to P. Jordan 3 February 1954 ([491 *], p. 442). Note that neither of these two eminent theoretical physicists discussed Thiry’s paper as regards its valuable content.

As to the field equations corresponding to (112 *) to (114 *), Thiry had calculated them in great detail with Cartan’s repère mobile for both a Euclidean or Lorentz metric of $V5$ , and also with a 5-dimensional matter tensor of the form of dust $ρu αuβ$ . From the 15th equation, he even had obtained “a new physical effect”: uncharged dust-matter could generate an electromagnetic field [[606 *], p. 79, footnote (1)]. He linked this effect to Blackett’s search for the magnetic field of a rotating body, described in Section 6.1.2. As to the interpretation of the fifteenth variable, the scalar field denoted by $V$ : for him $χTh := V 2G0$ with $G0 = 8πGNewton ∕c2$ was the gravitational coupling factor (“facteur de gravitation”) and put in front of the matter tensor [606 *], p. 72, 75, 77). In 1951, Y. Fourès-Bruhat proved existence and uniqueness theorems of “the unitary theory of Jordan–Thiry” [216 , 218 ].

11.1.1 Scientists working at the IHP on the Jordan–Thiry unified field theory

Further investigation of the “5-dimensional unitary theory of Jordan–Thiry” as it was named by Lichnerowicz’ and, later, by Thiry’s students, resulted in several doctoral theses. In one such by Françoise Hennequin-Guyon, after approximation methods parametrized by $-1 c2$ had been developed by her for the equations of motion in general relativity, they then were applied to Jordan–Thiry theory. A metric conformally related to the metric induced from the 5-dimensional space $V5$ was introduced such that the gravitational coupling function of the theory became a constant. The first approximation of a solution was then calculated [254 *]. Although stating that her interpretation via the conformal metric did agree with Jordan’s view (p. 77 of her thesis), surprisingly she did not mention his book in her bibliography. Like Y. Thiry, in the general formalism F. Hennequin also considered the “interior” case with a matter tensor by postulating field equation in $V5$

for “dust” (perfect fluid with zero pressure in 4 spatial dimensions): $Θ αβ = r νανβ, ναν α = − 1.r > 0$ later was called “pseudo-density”, $α ν$ is the 5-velocity. After the projection of the formalism into space-time, the electromagnetic and the scalar field, named $ξ$ , appeared. The conformal metric mentioned above is $ξds2$ with $ds2$ being the metric induced in space-time from $V 5$ . Electromagnetic field $F ij$ and induction $Hij$ are assumed to be related by $3 Hij = ξ Fij$ . Hennequin differed from Thiry by giving as ratio of charge and mass the expression $√ - 2 μρ = − c2∘βh--ξ2-, κ ρ = rξ(1 + hξ2) 1+ hξ2-$ with $h = u0$ being the timelike component of the 5-velocity and $β$ a constant ([254 ], p. 53), while Thiry had had $μ ∘-βh--- (ρ )T hi = 1+h2 ξ2$ ([606 *], p. 78). Her interpretation of the scalar field was also different from the one given by Lichnerowicz (cf. the end of Section 10.5).

In two subsequent theses of 1962 written by Robert Vallée and Pierre Pigeaud the defense of which was separated by only 4 months, Hennequin’s research within 5-dimensional theory was continued and generalized to charged perfect fluids. The study of such fluids had been begun by Roland Guy who extended Lichnerowicz’s results in relativistic hydrodynamics to “the formalism of the 5-dimensional unitary theories”. His matter tensor included “pressure” $p$ : $Θ = r ν ν − p γ ,γ αβ α β αβ αβ$ is the 5-dimensional Lorentz metric. His study of the congruences of the fluid’s streamlines comprised those with rotation: a “tenseur de tourbillon” $Ω αβ = ∂α(F νβ) − ∂β(F να)$ was introduced by R. Guy which after projection to $V4$ took the form:

Here, $μ$ stands for electric charge, $ϕi$ for the electromagnetic 4-potential, and $∫p -dp F = − exp p0ρ+p$ where $ρ$ is the mass density. P. Pigeaud used two metrics, the “natural” one $γ$ following from the projection of $R5$ to $R4$ , and another one conformal to it $ξγ$ . The first is employed for the calculation of the potentials (up to 2nd approximation), the 2nd for the study of the equations of motion. The reason is that for the “natural” metric uncharged test particles do not follow geodesics, yet for the conformal metric they do. At some point, Pigeaud had to make an ad hoc change of the energy-momentum tensor of matter (perfect fluid) not justified by empirical physics [495 *]. In a later development, Pigeaud did interpret the scalar fifteenth field variable as the field of a massive meson [387 , 496 ]. The investigations within Jordan–Thiry theory were carried on by Aline Surin-Parlange to the case of perfect fluid matter with an equation of state $ρ = f(p)$ where $ρ,p$ are mass density and pressure, f is an arbitrary function. The Cauchy problem for this case was solved, the existence of hydrodynamical waves shown, and their propagation velocity determined. A. Surin compared both, the “singularity” method and the method using the vanishing of the divergence of the matter tensor in 1st order approximation: they gave the same results. Unlike F. Hennequin who had assumed for the metric components²³⁷ that $γ04$ and $γ0A$ are of the same order $12 c$ , A. Surin assumed $γ04$ of order $-12 c$ , and $γ 0A$ of the order $1- c3$ . Unfortunately, as in previous work, the resulting equation of motion for charged particles still were in conflict with the classical electrodynamic equations. Surin thus had to change her choice of metric to a conformal metric in space-time; she found that the equations of motion in first approximation are independent of the conformal metric. Nevertheless, she had to admit that this did not help: “These last results, particularly those concerning the first approximation to the equations of motion seem to necessitate a modification of the field equations.”²³⁸ ([597 ], thèse , p. 126). A. Surin went on to calculate the 2nd approximation, an arduous task indeed, but which did not make the theory physically more acceptable. Moreover, in 2nd approximation, as in general relativity the results from both methods did not agree. Surin noted, that a modification of the field equations already had been suggested by H. Leutwyler [360 ], but did not comment on it. He had started from a variational principle mixing 5- and 4-dimensional quantities:
$∫ 1 4 (5)∘ ---(4)- δ dx ...dx R − g = 0$ , where $(5) R$ is the 5-dimensional curvature scalar expressible, due to the cylindricity condition, by quantities in space-time. He claimed to have removed the additional terms which precluded the derivation of the correct equations of motion. We will briefly present the further dissertation of R. Vallée below in Section 11.2.1. Huyen Dangvu also studied Jordan–Thiry theory and disagreed with its interpretation by Mariot & Pigeaud as a theory describing massive mesonic particles ([106 ], p. 4309). According to him:

“It seems preferable to us to interpret $ϕ$ as a scalar field without mass. Jordan–Thiry theory then will be a unitary theory of three massless fields: a scalar field, a vector field, and a 2-tensor field of spin 0, 1, 2, respectively.”²³⁹

11.1.2 Scalar-tensor theory in the 1960s and beyond

Around Y. Thiry in Paris, the study of his theory continued with J. Hély producing static spherically symmetric solutions of the field equations [253 , 252 , 251 ]. They describe a point particle with charge and mass and include the Schwarzschild metric. Due to the different field equations, a direct comparison with the earlier solution by Heckmann et al. of Jordan’s theory (cf. Section 3.1.2) seems not appropriate. However, H. Dangvu also contributed by looking at static and non-static spherically symmetric solutions to the Jordan–Thiry field equations in space-time.²⁴⁰ The solutions carry mass and some of them also electrical charge. Dangvu could compare them with those by Heckmann, Just, and Schücking in the Hamburg group around P. Jordan [108 *, 108 ].

From the full field (particle-) content of Kaluza–Klein theory, mainstream physics became interested most in the scalar field. In spite of the investigations within the framework of the ideas of Kaluza and Klein, and of Jordan’s approach within projective relativity, attention to the scalar field evolved in complete separation from its origin in unitary field theory. Soon, scalar-tensor theories were understood strictly as alternative theories of gravitation. We comment briefly on the loss of this historical perspective the more so as current publications on scalar-tensor theory are more interested in the subsequent modern developments than in the historical record [219 , 58 ]. In the Anglo-Saxon literature, scalar-tensor-theories run under the name of “Brans–Dicke theory”, or, at best, Brans–Dicke–Jordan theory, i.e., two authors being late with regard to Jordan and Thiry are given most of the credit; cf. standard references like ([242 ], pp. 59, 64, 71, 77, 362); ([438 ], pp. 1068, 1070, 1098; ([698 ], pp. 125, 126, 341).²⁴¹ True, the three groups of successive authors departed from different physical or mathematical ideas; Jordan from a varying gravitational constant as a hypothetical consequence of Dirac’s large number hypothesis, Thiry from a mathematical study of Kaluza–Klein theory and its global aspects, and Brans & Dicke from an implementation of their interpretation of Mach’s principle. A fourth author, W. Scherrer, must be included who was the first of all, and who considered the scalar field as a matter field coupled to gravitation; vid. Section 3.1.2 and [230 ]. Yet, one should keep in mind that, together with Yang–Mills theories and, perhaps non-local field theory, scalar-tensor theories of gravitation may be also considered as one surviving offspring of unitary field theory. Hence a total loss of historical memory with regard to the origins of scalar-tensor theory appears unjustified.²⁴²

11.2 6- and 8-dimensional theories

Since Kaluza had proposed a 5-dimensional space as a framework for a unitary field theory of the gravitational and electromagnetic fields in 1919, both experimental elementary particle physics and quantum field theory had progressed. Despite the difficulties with divergencies, since the mid 1950s renormalization procedures had been stable enough to make quantum field theory acceptable and needed as a proper formalism for dealing with the known elementary particles. Nevertheless, in some approaches to unified field theory, it still was thought useful to investigate classical theory. Thus, in this context theories with new degrees of freedom for the new fields ( $π$ - and $μ$ -mesons, neutrino) had to be constructed. We noticed that both in the Einstein–Schrödinger affine field theory and in the Jordan–Thiry extension of Kaluza’s theory such attempts had been made. The increase in the dimension of space seemed to be a handy method to include additional fields. In Section 7.2.2 of Part I, papers of Rumer, Mandel, and Zaycoff concerning 6-dimensional space have been mentioned. None of them is referred to by the research described below. It involved theorists working independently in the USA, Great Britain, and France.

11.2.1 6-dimensional theories

The first of these, B. Hoffmann in Princeton, wanted to describe particles with both electric charge $e$ and magnetic charge $μ$ .²⁴³ Because the paths of electrically charged particles could be described as geodesics of Kaluza’s 5-dimensional space, he added another space-dimension. The demand now was that his particles with both kinds of charge follow geodesics in a 6-dimensional Riemannian (Lorentzian) space $R6$ . [277 *]. The coordinates in $R6$ are denoted here by $A,B = 0,1, 2,3,4,∞$ , in $5 R$ by $α, β,⋅⋅⋅ = 0,1,...,4$ , and in space-time $i,j,= 1,2, 3,4$ . The metric $kAB$ of $6 R$ contains, besides the metric $gij$ of space-time, two vector fields $kk0 = ϕk,k∞j = ψk$ and three scalar fields $k00,k0∞, k∞,∞$ . The scalar fields are disposed of immediately: $k00 = 1$ (cylinder condition), $k0∞ = 0,k∞,∞ = − 1$ , while $ϕk$ is taken to be the electrical 4-potential: $∂ϕi ∂ϕj ϕij = 1∕2(∂xj − ∂xi)$ and $ψk$ the corresponding quantity following from the dual of the electrical field tensor $ij ∗ij √ --- −1 ijrs ψ = ϕ = 1∕2( − g) 𝜖 ψrs$ .

The geodesic equation $d2x2A + KABC dxB-dxC = 0 ds ds ds$ with the Levi-Civita connection $KABC$ is decomposed into 3 groups corresponding to $d2xk d2x0- d2x∞-- ds2 ..., ds2 ..., ds2 ...$ . From the last two equations follows $C ϕC dxds = const. != − e∕2m$ and $C ψC dxds = const.=!− μ∕2m$ such that the projection of the geodesic equation of $6 R$ into space-time reads as: $d2xk k dxrdxs e- k dxr- k-μdxr ds2 + {rs}ds ds − mϕ r ds − ψ rm ds = 0$ . In order that the geodesics are timelike curves, the condition $∂ϕp ∂ϕq ∂ψp ∂ψq gmpgnq (∂xq − ∂xp-) = 2√1−g𝜖mnpq(∂xq − ∂xp)$ must be fulfilled. It does as a consequence of the field equations derived from:

and which correspond merely to Maxwell’s equations and to “the corresponding field equations of the projective theory”( [277 ], p. 29). The curvature scalar of $R6$ is $C = R − ϕrsϕsr − ψrsψsr$ with $R$ being the curvature scalar of space-time. With topological questions not asked by him, Hoffmann’s excursion into 6-dimensional space for describing the path of a magnetic monopole was entirely unnecessary. In fact, in his next paper he applied the method of Einstein and Mayer [175 ] presented in Section 6.3.2 of Part I to his particle with electric and magnetic charge and developed a theory in 4 dimensions [276 ].

In Manchester, where A. Papapetrou and P. M. S. Blackett worked on possible empirical consequences of UFT (cf. Section 6.1.2), while L. Rosenfeld wrote about the quantum theory of nuclear forces, a research fellow Julius Podolanski embarked on a unified field theory in six dimensions [498 *]. His motivation came from Dirac’s equation, more precisely from the algebra of the 15 $γ$ -matrices, interpreted as representing the algebra of rotations in a 6-dimensional Riemannian space. His manifold $V 6$ is decomposed into a flat and geodesic 2-dimensional “sheet” generated by a spacelike and a timelike translation²⁴⁴ , and a 4-dimensional Lorentz space $V4$ (identified with space-time), both normal to each other. Likewise, any 6-vector can be decomposed into a part in the 2-sheet and a 4-vector. A material point particle is assumed to follow a geodesic of $V6$ . From the connection of $V6$ , an induced connection in $V4$ can be obtained. Invariance with regard to the translations is called “gauge”-invariance in $V4$ . This means that gauge-invariant quantities do not depend on the sheet-coordinates. It turned out that the metric of $V6$ , besides leading to the 4-dimensional metric $gij$ , introduced two skew-symmetric tensors named fields of constraint. One of them is interpreted as the electromagnetic field. The interpretation of the second skew field is left open. The scalars in the theory (norms and inner product of the translations) are assumed constant. The field equations were derived from the curvature scalar of $V6$ and obviously lead to Einstein’s vacuum field equations in six dimensions; a matter tensor in $V6$ is inserted by hand and restricted to charged pressureless matter. After a projection into $V 4$ , it turned out that the currents and energy-momentum tensors of the two skew-symmetric fields had different signs: One field has a positive, the other a negative energy content. This was not seen as unphysical by the author; he commented: “So it seems that this theory may contribute to the problem of self-energy”, an allusion to the divergence problem in quantum field theory ([498 *], p. 235), and to the fact that the energy density can become negative in quantum field theory. But he was disappointed that the 2nd skew field also had infinite reach: its quanta are massless. He tried to mend this by demanding that a particle follow a lightlike geodesic in $V6$ , but did not really succeed. His conclusion was: “A unitary field theory does not seem possible without introducing quantum theoretical concepts. This – classical – treatment therefore can only be incomplete” ([498 *], p. 258). Pauli noticed this paper; in the summer term 1953, he had given a course on general relativity including Kaluza’s 5-dimensional theory. In a letter to Fierz of 3 July 1953 referring to this course we can read:

“In it, also Kaluza’s 5th dimension did occur, and per se it is quite satisfying if now, in place of it, two additional dimensions with the 3-dimensional rotation group are introduced (this has already been suggested, if only formalistic, by e.g., Podolanski who has a 6-dimensional space).” ([491 *], p. 186–187.) ²⁴⁵

Pauli’s interest really went toward a paper of Pais who had suggested a 6-dimensional $ω$ -space consisting of space-time and an internal sphere-space affixed at any point [468 ]. From here, a direct line leads to Pauli’s (unpublished) derivation of the non-Abelian $SU (2)$ gauge theory presented in letters to Pais. For a detailed history cf. O’Raifeartaigh’s book ([463 ], Chapter 7). cf. also the section “A vision of gauge field theory” in [194 ], pp. 476–488.

When Josette Renaudie wrote her dissertation with A. Lichnerowicz and M.-A. Tonnelat on 6-dimensional classical unitary field theory in 1956, she also used elementary particles as her motivation [505 , 506 *], ([507 ], p. 3; [508 ]). However, unlike Podolanski [498 ] and Yano & Ogane [715 ], she worked in a 6-dimensional Lorentz space: 2 space-dimensions are added. While Yano & Ogane employed a general 2-parameter isometry group and claimed to have the most general formalism, Renaudie admitted a general Abelian 2-dimensional isometry group, thus keeping 3 arbitrary scalar functions. Her two Killing vectors with regard to which the projection from 6-dimensional $V6$ along the trajectories of one Killing vector first to to $V5$ , and then with the 2nd Killing vector to space-time is performed, are spacelike. Again, an Einstein field equation $SAB = PAB, (A, B = 0,1,...,5)$ is written down with the “matter tensor” $PAB$ getting its interpretation backwards from the corresponding 4- and 5-dimensional quantities. The Einstein tensor now has 21 independent components and can be split into a rank 2 symmetric tensor, two 4-vectors and 3 scalar functions. Renaudie gave two interpretations: (1) these variables stand for a hyperfield composed of gravitational, electromagnetic and mesonic fields (with the mesonic field a complex vector field), and (2) the field of a particle of maximal spin 1 in interaction with the gravitational field (p. 68/69). Note that in both interpretations the scalar fields remain unrelated to physical quantities. The terms in the field equations describing the interaction of the mesonic and electromagnetic fields are independent of the geometrical objects in space-time. The Cauchy initial value problem can be set up and solved properly.²⁴⁶

Five years later, another thesis advised by A. Lichnerowicz continued the work of J. Renaudie: R. Vallée investigated “The relativistic representation of charged perfect fluids” within 6- and 5-dimensional “space-times” [666 *]. According to its title, a focus lies in the material content, a perfect fluid, which also had been briefly dealt with by J. Renaudie ([506 ], cf. her Section 34). Vallée’s ansatz for the matter tensor in six dimensions is taken over from space-time:

where $Γ IJ$ is the metric in 6-dimensional space $6 V$ with Lorentz signature, $WI$ with $I J W W Γ IJ = 1$ , the 6-velocity tangent to the streamlines, and $P > 0,Π > 0$ denote “density” and “pressure” ([666 ], p. 36). An equation of state $P + Π = f(Π )$ is assumed in order to get the corresponding relation in space-time. Projection into space-time (metric $g ij$ ) via $V 5$ (metric $γ αβ$ ) led to the matter tensor of a charged fluid of the form $′ ρuiuj + τij$ with the electromagnetic energy-momentum tensor: $′ 1 r r 1 rs τij = 2(H iFjr + Fi Hjr ) − 4gijHrsF$ . Electromagnetic field $Fij$ and induction field $Hij$ were assumed to be related through $Hij = ξ3Fij$ where $ξ2 = − γ00$ stands for the additional scalar field in Jordan–Thiry theory. As with Renaudie, $V 6$ admits a 2-parameter Abelian group of global isometries. This is research extending the work of Renaudie to further classes of fluids; it fits to the doctoral theses described in Section 11.1.1 – in essence, they presented formal developments with restricted relevance to the envisaged physics of unitary field theory.

In 1963, Mariot & Pigeaud again took up the 6-dimensional theory in a paper²⁴⁷ [388 ]. After the introduction of a conformal metric in $∗ V4, gij = ηξgij, i,j = 1,2,3,4$ with $2 2 ξ ,η$ being the norms squared of the 2 Killing vectors from the additional dimensions, they studied the linear approximation of the theory with incoherent matter $muiuj$ . They were able to identify in $V4$ matter tensors belonging to the electromagnetic field, to a neutral vector meson field and to both a massive neutral and a charged scalar meson field. Yet, some remaining terms were still not amenable to a physical interpretation.

11.2.2 Eight dimensions and hypercomplex geometry

As early as 1934, an eight-dimensional space with two time-dimensions was introduced in order to describe the gravitational field corresponding to an accelerated electromagnetic field. It then turned out that the two time coordinates were related by the eight geodesic equations such that, essentially, a 7-dimensional Lorentz space remained. Einstein’s vacuum field equations in 8 dimensional space were assumed to hold [421 , 422 ]. In view of the substantial input, the results, reached by approximate calculations only, were meagre: an approximate solution of the gravitational two-body problem; only static electric and constant magnetic fields could be described.

In the 1950s, the idea of using an 8-dimensional space as the stage for UFT seemingly arose by an extension of the mapping of connections in the same 4-dimensional space to a second copy of such a space. In Section 2.1.2 we have noted F. Maurer-Tison’s interpretation of (30 *): The transport of a covariant vector $νi$ with regard to the connection $L&tidle; k ij$ corresponds to the transport of the contravariant vector $→k ν = gksνs$ with regard to $Lijk$ . This situation was turned into a geometry named semi-metric by Pierre-V. Grosjean in which two identical 4-dimensional spaces (“distinct universes”) were introduced with the connection $L$ acting in one and its Hermitian conjugated connection $&tidle;L$ in the other [235 ]. Inner products of vectors were allowed only if one vector is in the one space, the other in the second. His conclusion that the semi-metric geometry would be “the key to the unitary theory, in the same way as metric geometry was the key to general relativity” is more than overbearing. All that remains is a (physically empty) mathematical formalism the only consequence of which was to eventually motivate the scheme of applied mathematics to be discussed next. As mentioned in Section 2.6, in close parallel to the complex numbers, a ring formed from $x + 𝜖y$ with real numbers $x,y$ and $𝜖 × 𝜖 = 1$ can be introduced: the hypercomplex numbers. Albert Crumeyrolle built a unitary field theory of the Einstein–Schrödinger type on this number field [96 , 97 *, 98 , 95 ]. In a generalization of the formalism noted in Section 10.5.4, he now introduced a 2n-dimensional $∞ C$ -space $2n V$ and local charts around points by assigning n hypercomplex numbers $zα = xα + 𝜖xα∗, ¯z = xα − 𝜖xα∗, α,α ∗ = 1,2,...,n$ . In another basis, $zα = 1(ξ α + ξα ∗) + 𝜖 1(ξα − ξα∗) 2 2$ . The $xα,(ξα),xα ∗,(ξα∗)$ transform among themselves. Again, a “natural adapted reference system” was introduced in which dual bases derive one from the other by the regular and diagonal $2n × 2n$ -matrix in $V 2n$ :

( ) A 0 . 0 ¯A

Here, $A, ¯A$ are $n × n$ -matrices with $α′ α′ α′ α′ α′ α′ A = ∂z∂zλ-= ∂∂xxλ-+ 𝜖 ∂∂xxλ∗,A¯= ∂∂z¯zλ-= ∂∂xxλ-− 𝜖 ∂∂xxλ∗.$ In $V 2n$ , locally a real symmetric covariant tensor could be introduced which decomposes in the coordinate system $(zα,z α∗)$ into parts $γ ,γ ,γ αβ α∗β∗ αβ∗$ . Likewise connections and special unitary connections were defined. In the end, a system generalizing the weak field equations of UFT resulted. For the physicist, and the more so for the historian of physics, the paper is a maze of mathematical structures. The author’s statement that “a new field can be introduced the physical signification of which is not examined here but which could perhaps have something to do with the hypothesis of an inertial effect of the spin”, remains unfounded ([97 ], p. 105).

	Abstract
1	Introduction
2	Mathematical Preliminaries
	2.1	Metrical structure
	2.2	Symmetries
	2.3	Affine geometry
	2.4	Differential forms
	2.5	Classification of geometries
	2.6	Number fields
3	Interlude: Meanderings – UFT in the late 1930s and the 1940s
	3.1	Projective and conformal relativity theory
	3.2	Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
	3.3	Non-local fields
4	Unified Field Theory and Quantum Mechanics
	4.1	The impact of Schrödinger’s and Dirac’s equations
	4.2	Other approaches
	4.3	Wave geometry
5	Born–Infeld Theory
6	Affine Geometry: Schrödinger as an Ardent Player
	6.1	A unitary theory of physical fields
	6.2	Semi-symmetric connection
7	Mixed Geometry: Einstein’s New Attempt
	7.1	Formal and physical motivation
	7.2	Einstein 1945
	7.3	Einstein–Straus 1946 and the weak field equations
8	Schrödinger II: Arbitrary Affine Connection
	8.1	Schrödinger’s debacle
	8.2	Recovery
	8.3	First exact solutions
9	Einstein II: From 1948 on
	9.1	A period of undecidedness (1949/50)
	9.2	Einstein 1950
	9.3	Einstein 1953
	9.4	Einstein 1954/55
	9.5	Reactions to Einstein–Kaufman
	9.6	More exact solutions
	9.7	Interpretative problems
	9.8	The role of additional symmetries
10	Einstein–Schrödinger Theory in Paris
	10.1	Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
	10.2	Tonnelat’s research on UFT in 1946 – 1952
	10.3	Some further developments
	10.4	Further work on unified field theory around M.-A. Tonnelat
	10.5	Research by and around André Lichnerowicz
11	Higher-Dimensional Theories Generalizing Kaluza’s
	11.1	5-dimensional theories: Jordan–Thiry theory
	11.2	6- and 8-dimensional theories
12	Further Contributions from the United States
	12.1	Eisenhart in Princeton
	12.2	Hlavatý at Indiana University
	12.3	Other contributions
13	Research in other English Speaking Countries
	13.1	England and elsewhere
	13.2	Australia
	13.3	India
14	Additional Contributions from Japan
15	Research in Italy
	15.1	Introduction
	15.2	Approximative study of field equations
	15.3	Equations of motion for point particles
16	The Move Away from Einstein–Schrödinger Theory and UFT
	16.1	Theories of gravitation and electricity in Minkowski space
	16.2	Linear theory and quantization
	16.3	Linear theory and spin-1/2-particles
	16.4	Quantization of Einstein–Schrödinger theory?
17	Alternative Geometries
	17.1	Lyra geometry
	17.2	Finsler geometry and unified field theory
18	Mutual Influence and Interaction of Research Groups
	18.1	Sociology of science
	18.2	After 1945: an international research effort
19	On the Conceptual and Methodic Structure of Unified Field Theory
	19.1	General issues
	19.2	Observations on psychological and philosophical positions
20	Concluding Comment
	Acknowledgements
	References
	Footnotes
	Biographies